Some observations on the Green function for the ball in the fractional Laplace framework

TL;DR

This paper provides an accessible, elementary exposition of the Green function for the fractional Laplace equation on a ball, aiming to aid students and educators in understanding this mathematical concept without complex methods.

Contribution

It offers a clear, self-contained proof of the Green function representation for the fractional Laplace operator on a ball, using only elementary calculus techniques.

Findings

Elementary proof of Green function representation

Accessible exposition suitable for educational purposes

No advanced probabilistic or algebraic methods used

Abstract

We consider a fractional Laplace equation and we give a self-contained elementary exposition of the representation formula for the Green function on the ball. In this exposition, only elementary calculus techniques will be used, in particular, no probabilistic methods or computer assisted algebraic manipulations are needed. The main result in itself is not new, however we believe that the exposition is original and easy to follow, hence we hope that this paper will be accessible to a wide audience of young researchers and graduate students that want to approach the subject, and even to professors that would like to present a complete proof in a PhD or Master Degree course.